# Properties of Natural Numbers

Properties of Natural Numbers
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 1 Introduction to Natural Numbers 2 Closure Property 3 Associative Property 4 Commutative Property 5 Distributive Property 6 Solved Examples 7 Challenging Questions 8 Practice Questions 9 Maths Olympiad Sample Papers 10 Frequently Asked Questions (FAQs)

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## Introduction

Natural numbers are the numbers that are used for counting.

The set of natural numbers in Mathematics is the set $$\textbf{ \{1,2,3,...\} }$$.

The set of natural numbers is denoted by the symbol, $$N$$.

 N = $$\{1,2,3,4,5,...\}$$

The four properties of natural numbers are as follows:

1. Closure Property
2. Associative Property
3. Commutative Property
4. Distributive Property

Let's explore them in details.

## 1. Closure Property

• Add any two natural numbers and you will see that the sum is again a natural number.

For example,

\begin{align}1+2 = 3\end{align}

Here, $$3$$ is a natural number.

• In the same way, multiply any two natural numbers and you will see that the product is again a natural number.

For example,

\begin{align}3 \times 2 = 6\end{align}

Here, $$6$$ is a natural number.

So the set of natural numbers, $$N$$ is closed under addition and multiplication.

So the closure property of $$N$$ is stated as follows:

For all $$a,b \in N$$

 $a + b \in N\\ {\text{ and }}\\a \times b \in N$

You can go ahead entering any two natural numbers in the simulation below to see their sum and the product.

The sum and product are always natural numbers! Interesting, isn’t it?

## 2. Associative Property

The sum or product of any three natural numbers remains the same though the grouping of numbers is changed.

Example 1:

$(1+2)+3 = 1+(2+3)$

because

\begin{align} (1+2)+3&=3+3=6 \\1+(2+3)&=1+5=6\end{align}

Example 2:

$(1\times 2) \times 3 = 1 \times (2 \times 3)$

because

\begin{align} (1\times 2) \times 3&= 2 \times 3 =6 \\1 \times (2 \times 3) &=1 \times 6 =6\end{align}

So the set of natural numbers, $$N$$ is associative under addition and multiplication.

So the associative property of $$N$$ is stated as follows:

For all $$a,b,c \in N$$

 $\begin{gathered} a + \left( {b + c} \right) = \left( {a + b} \right) + c \\ {\text{and}} \\ a \times \left( {b \times c} \right) = \left( {a \times b} \right) \times c \\ \end{gathered}$

You can go ahead entering any three natural numbers in the below simulation and see their sum and the product can be found in two ways.

The sum and product are NOT changed even when the grouping of numbers is changed. Have you noticed this?

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## 3. Commutative Property

The sum or product of two natural numbers remains the same even after interchanging the order of the numbers.

Example 1:

$2 +3=3+2$

because

\begin{align} 2+3&=5\\3+2&=5\end{align}

Example 2:

$2\times 3= 3 \times 2$

because

\begin{align} 2 \times 3 &=6\\3 \times 2 &=6\end{align}

So the set of natural numbers, $$N$$ is commutative under addition and multiplication.

So the commutative property of $$N$$ is stated as follows:

For all $$a,b \in N$$

 $\begin{gathered} a + b = b + a \\ {\text{and}} \\ a \times b = b \times a \\ \end{gathered}$

You can go ahead entering any two natural numbers in the simulation here and see their sum and the product found in two ways.

The sum and product are NOT changed even when the numbers are interchanged. Have you observed this?

Let us summarise these three properties of natural numbers in a table.

Operation Closure Property Associative Property Commutative Property
Subtraction no no no
Multiplication yes yes yes
Division no no no

## 4. Distributive Property

The distributive property of multiplication over addition is

 \begin{align}a \times (b+c) &= a \times b + a \times c\end{align}

Example: $3 \times (2+5) = 3 \times 2 + 3 \times 5$

because

\begin{align}\ 3 \times (2+5) &= 3 \times 7 =21 \\3 \times 2 + 3 \times 5 &=6+15=21\end{align}

The distributive property of multiplication over subtraction is

 \begin{align}a \times (b-c) = a \times b - a \times c \end{align}

Example: $3 \times (2-5) = 3 \times 2 - 3 \times 5$

because

\begin{align} 3 \times (2-5) &= 3 \times (-3) =-9 \\3 \times 2 - 3 \times 5 &=6-15=-9\end{align}

Think Tank
1. Is $$N$$ closed under subtraction and division?
2. Is $$N$$ associative under subtraction and division?
3. Is $$N$$ commutative under subtraction and division?

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## Solved Examples

 Example 1

The set of natural numbers is closed under which of the operations?

2. Subtraction
3. Multiplication
4. Division

Solution:

If we assume any two natural numbers, their sum and the product are also the natural numbers.

But their product and quotient don't need to be the natural numbers.

For example, $$1$$ and $$2$$ are natural numbers.

$1-2=-1\\1 \div 2 =0.5$

Here, the difference and the quotient are NOT natural numbers.

So the set of natural numbers is closed only under addition and multiplication.

 Options 1 and 3
 Example 2

Find the following product using the distributive property.

$62 \times 35$

Solution:

By using the distributive property, we can write the given product as follows:

\begin{align}62 \!\times\! 35 &\!=\!60\!\times\! 30 \!+\! 60\!\times\! 5 \!+\! 2 \!\times\! 30 \!+\! 2 \!\times\! 5 \\[0.3cm]&\!=\!-1800\!+\!300\!+\!60\!+\!10\\[0.3cm] &\!=\!2170\end{align}

 $62 \times 35 =2170$
 Example 3

$$13 \times (12 \times 15) = (13 \times 12)\times 15$$ is an example of ____________ property.

a. Closure property under multiplication

b. Associative property under multiplication

c. Commutative property under multiplication

Solution:

The associative property under multiplication is:

$a\times (b \times c) = (a \times b) \times c$

So the given equation is an example of "associative property under multiplication".

 Option b

Challenging Questions
1. Find the product using the distributive property:
$35 \times 77$
2. The set of natural numbers is commutative under which of the operations?

 (a) Addition (b) Subtraction (c) Multiplication (d) Division

## Practice Questions

Here are a few problems related to the properties of natural numbers.

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

## 1. What are the properties of natural numbers?

The properties of natural numbers are:

1. Closure property
2. Associative property
3. Commutative property
4. Distributive property

## 2. Is the set of natural numbers associative under division?

The set of natural numbers is NOT associative under division.

For example, let us consider three natural numbers $$6, 4$$ and $$2$$.

Then:

$(6\!\div\! 4) \!\div\! 2 \!=\! 3 \!\div\! 2 \!=\! 1.5 \\ 6 \!\div\! (4 \!\div\! 2) \!=\! 6 \!\div\! 2 \!=\! 3$

Thus,

\begin{align}(6\!\div\! 4) \!\div\! 2 \!\neq\! 6 \!\div\! (4 \!\div\! 2) \!=\! 6 \!\div\! 2 \!=\! 3\end{align}

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